1. Fringe-Fitting: Correcting for delays and rates

2. Summary
Radio signals detected by an interferometer are affected by a delay
Wavefront of signal will arrive at each antenna at a different time (i.e. different delays)
Correlator corrects for the changes in the delays, but the model will have errors
Due to the large distances, this error changes significantly with time and frequency
Corrected using the Fringe-Fitting technique

3. Introduction Let us start with a simple 2 element radio interferometer
Wavefronts of a signal from a distant source, arrives at one antenna with a geometrical delay,τobs = D/c sinθ
Φ= 2πντobs (interferometer phase)
τobs changes with time → fringe rates

4. Introduction Let us start with a simple 2 element radio interferometer
Signals from both antennas are combined in a correlator
Correlator estimates and corrects for τobs
For connected arrays e.g. JVLA, ATCA, KAT-7 this simple geometrical delay is sufficient … not so for VLBI

5. VLBI vs shorter-BI Just interferometry
Longer baselines (100's to 1000's km)
→ very high resolution
→ sensitive to compact sources with high surface brightness
→ increased time and bandwidth smearing
→ less data averaging and larger data size for same FOV
Independent clocks and equipment → phase/delay errors
The delay and rate of the wavefronts are affected by different effects
Must be estimated and removed during correlation

6. VLBI Geometric Model Terms that affect the delay delay > few cm
Most radio astronomers don't have to worry about these effects
The most dominant would be the atmosphere
if not corrected for will introduce large pase errors and decorrelate the signals
Did you know?: by observing very distant bright compact Quasars, VLBI can monitor the changes due to many of these effects … including the motion of the tectonic plates!
Table from Walker 1999, Ch. 22, the White Book

7. Residual Delay & Rate Errors
Correlator model isn't perfect
Residual phase, delay and rate errors
Mainly from atmospheric fluctuations, clock errors
Recall the interferometer phase → Φ= 2πντobs
τobs = τg + τstr + τtrop + τiono + τinstr + … + εnoise
Phase error will depend on the delay error
Linear phase model (first order expansion):
Δφ 𝑡,ν = φ 0 + δφ δν Δν+ δφ δ𝑡 Δ𝑡
Phase error at reference time and freq + delay + delay rate
Some cases (e.g. space VLBI, mm-VLBI ??) may require higher orders
Errors are corrected by Fringe-Fitting

8. Fringe-Fitting
Sources of delay and rate errors can be separated into contributions from each antenna
Baseline depndent errors → difference of antenna dependent errors
Phase errors for baseline i,j
Fringe-fitting involves solving the above equation, to obtain the errors
Via observations of a bright calibrator → phase referencing
Assumes source is a point source at the phase centre
Can be done per baseline or global (i.e. combine all baselines)
Without, cannot average in phase and time
Worse for weaker targets

9. Baseline Fringe-Fitting
FT to the delay-rate domain (data is in t-ν domain)
FT each baseline independently
Peak in delay-rate domain, gives the error for that baseline
Must detect the source on all baselines
Does not maintain antenna based or closure relationships
Not useful for weaker sources

10. Global Fringe-Fitting
Use all baselines to jointly estimate the antenna phase, delay and rate relative to a reference antenna
Solves the baseline phase error equation, with one of the antennas set to the reference antenna
Delay, rate and phase residuals for reference antenna are set to zero
Hence only measures difference, not absolute errors
Assumes calibrator is a bright point source at phase center
Similar to self-calibration as source structure is part of the model
Implemented in AIPS

11. Global Fringe-Fitting
Random atmospheric phase fluctuations
Different for each antenna
Baseline phase error equation only adequate for a limited time → coherence time
Solution interval:
> coherence time: increases sensitivity, but decreases number of possible solutions and increases phase ambiguities
< coherence time: decreases sensitivity, increases number possible of solutions, decreases phase ambiguities
ideal: long enough to have high SNR, but short enough to decrease phase ambiguities
Good starting: set solutions interval to have one solution per phase reference cycle

12. Phase Referencing
Fringe-fitting requires observations of a bright, compact source → the phase-calibrator, C.
Nodding between C and target (T)
Cycle time must be shorter than the atmospheric fluctuations
~10 mins at 5 GHz; ~5 mins at 1.6 GHz
C must be close to T (< 1 deg)
Antenna positions must be known to within a few cm!
Obtain solutions of the phase, rate and delay by applying the Fringe-fitting technique to C and interpolate to T
Biggest problems
- wet troposhere & fewer calibrators at high freq
- Ionosphere at low freq

13. Closure Phase
All antennas have different random phase fluctuations due to atmosphere
Closure phase, φc is the sum of simultaneously observed phases of a source on three baselines forming a triangle
φc = φ12 + φ23 + φ31 + noise
Independent of station based phase errors
phase errors due to different atmospheric variations are cancelled in the closed loop
Fringe-fitting (and self-cal) uses this triangle to solve for the residual phases, rates and delay
Original ref: Rogers et al 1974, ApJ, 193, 293
Phase contributions from each baseline =
true phase + (difference between the random atm phases)
12->
23->
31->
Summing the total phases from each baseline, the phases from the atm cancels:
Φ 12 = φ 12 + θ 1 − θ 2
Φ 23 = φ 23 + θ 2 − θ 3
Φ 31 = φ 31 + θ 3 − θ 1
φ 𝑐 = Φ 12 + Φ 23 Φ 31 +𝑛𝑜𝑖𝑠𝑒
φ 𝑐 = φ 12 + φ 23 + φ 31 +𝑛𝑜𝑖𝑠𝑒

14. Fringe-Fitting in practice
Currently, fringe-fitting is implemented in the Astronomy Imaging Package (AIPS) via the task FRING
AIPS was the standard radio reduction and imaging software
Mostly used by VLBI astronomers (because of fringe-fitting) and people stuck in their old ways (like me!)
FRING is currently being written (or copied) for CASA (the new software)
Until then you will need to use AIPS ...
Typical AIPS User

15. Fringe-Fitting in practice
Currently, fringe-fitting is implemented in the Astronomy Imaging Package (AIPS) via the task FRING
AIPS was the standard radio reduction and imaging software
Mostly used by VLBI astronomers (because of fringe-fitting) and people stuck in their old ways (like me!)
FRING is currently being written (or copied) for CASA (the new software)
Until then you will need to use AIPS …
Typical AIPS User

16. AIPS AIPS is written by the NRAO (US)
ver 1 → 1978 in fortran (the user inteface hasn't changed in 38 years!)
Composed of Tasks that are called in the terminal
The user input values (or string commands) to keywords called parameters
Image/plot viewer
User input terminal
Message window

17. UVFITS UVFITS - AIPS input data format
Composed of header tables and data columns
Each table has records of specifc information (e.g.):
FQ: Frequency info; SU: Source position observed; AN: antenna info
AIPS do not edit the data, but create tables that is appended (e.g.):
FG table: Flag tables (records of bad data to delete)
SN table: Calibration solutions obtained from calibrators
CL table: Calibration solutions applied to the targets
BP: instrument bandpass
The information provided by these tables are only applied at the end of the calibration process via the task 'SPLIT'

18. Fringe-Fitting in practice
Fringe-fitting will be done after amplitude calibration and correcting for instrumental delays (due to different path lengths of the various IFs)
eg: 10 ant 1.6GHz. Obs with 8 x 16 x 1MHz channels
Phase vs frequecy on baselines to JB (the Lovell telescope)
unresolved source: J
before Fring-fitting phase variations J

19. Fringe-Fitting in practice
Fringe-fitting will be done after amplitude calibration and correcting for instrumental delays (due to different path lengths of the various IFs)
eg: 10 ant 1.6GHz. Obs with 8 x 16 x 1MHz channels
Phase vs frequency post fringe-fitting
Phases zero +/1 noise for point source at phase centre
Performed on each calibrator and applied to targets
Reference antenna: Effelsberg (100m)
Solutions interval = to one calibrator scan
Note: this is Global Fringe-Fitting

20. Summary
VLBI observations, due to an imperfect delay model will introduce residual phase, delay and rate errors
These errors, changes with time and frequency and must be removed before imaging
Discussed the fringe-fitting technique and explained phase referencing
Quickly explained the practical application in AIPS